A vastly improved performance by the home side, separated by 31 places from their London opponents, ended with Forest having their hearts broken in the 86th minutes when Radzinski equalised after Marlon King had put the home side in front deep into the second half.
Andy Reid had scrambled the ball through two Fulham defenders in the 72nd minute and King pounced on the opportunity to roll the ball over the line despite Mark Crossley's best efforts.
Forest's joy lasted only 12 minutes though as Luis Boa Morte neatly squeezed the ball to Radzinski in the Forest box, and the Canadian shot under the onrushing Paul Gerrard from six yards out to take the game into extra-time.
In the 93rd minute Radzinski then scored his second goal of the game as his 18-yard diagonal shot found the top corner of Gerrard's goal.
Eight minutes later substitute McBride thought he had finally booked his team's place in the quarter-finals as he made the score 3-1 with a 15-yard shot.
But three minutes later Reid reduced the deficit to just one goal when he slotted the ball into the net after Crossley had only managed to block Paul Evans' initial shot from the edge of the box.
Cole finally made the game safe two minutes before the final whistle when he lobbed the ball over Gerrard to make the score 4-2.